By I. A Barnett

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Q} to {1, . . , t}, so there are again at most nq possibilities for H. 9). Suppose finally that G is finite. If H ≤ G and H ∩ N = D then H/D ∼ = HN/N ≤ Q so H is generated by D together with at most rk(Q) further elements. 11) follows. In part (iii) of the last proposition we bounded s(G) in terms of s(N ) and rk(Q). 3 Suppose that G is finite and that N is soluble, of derived length l and rank r. Then 3r 2 +lr s(G) ≤ s(Q) |N | lr 3r 2 +lr |Q| ≤ s(Q) |G| . 4 If Q is finite and A is a finite Q-module of (additive) rank r then 3r 2 −1 r H 1 (Q, A) < |A| |Q| .

It follows that for each i, Y1vi Y vi , and hence by the preceding vk vi v1 remark Y1 Y . Therefore Y = Y1 · . . · Y1 has order at most sk , so it has exactly this order and φ|Y : Y → N is an isomorphism. Also, as Y is finite the relations Y xi ≤ Y ensure that Y is normal in X. Putting every wi and wi equal to 1 in the above presentation we obtain a presentation of X/Y , which is just the given presentation x ; u1 , . . , ur for G/N . Hence φ also induces an isomorphism from X/Y onto G/N . Thus φ is injective and X ∼ = G as claimed.

GROWTH TYPES 33 In particular we have |Hom(A, A)| | |Hom(A, Q1 ⊕ · · · ⊕ Qk )| = |Hom(Q1 ⊕ · · · ⊕ Qk , A)| | |Hom(Q1 ⊕ · · · ⊕ Qk , Q1 ⊕ · · · ⊕ Qk )| . 6, k k |End(Qi )| . 11 Growth types To conclude the chapter, we interpret some of the results in terms of growth type. Recall that a group G is said to have subgroup growth of type f if there exist positive constants a and b such that sn (G) ≤ f (n)a for all n sn (G) ≥ f (n)b for infinitely many n; and that G has growth of strict type f if the second inequality holds for all large n.